Tarski–Mostowski–Robinson theorem
E353626
The Tarski–Mostowski–Robinson theorem is a fundamental result in model theory that characterizes when a class of structures is first-order axiomatizable, linking definability properties with closure under ultraproducts and isomorphisms.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Tarski–Mostowski–Robinson characterization of elementary classes | 1 |
| Tarski–Mostowski–Robinson theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3380882 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Tarski–Mostowski–Robinson theorem Context triple: [Tarski's undefinability theorem, relatedTo, Tarski–Mostowski–Robinson theorem]
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A.
Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
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B.
Herbrand's theorem
Herbrand's theorem is a fundamental result in mathematical logic and proof theory that characterizes the validity of first-order formulas via finite sets of ground instances, forming a basis for automated theorem proving.
-
C.
Hilbert’s program
Hilbert’s program was an influential early-20th-century initiative in the foundations of mathematics that sought to formalize all of mathematics and prove its consistency using finitistic methods.
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D.
Recherches sur la théorie de la démonstration
Recherches sur la théorie de la démonstration is Jacques Herbrand’s foundational work in mathematical logic, introducing key results in proof theory and what is now known as Herbrand’s theorem.
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E.
Hamilton’s compactness theorem
Hamilton’s compactness theorem is a fundamental result in geometric analysis that provides conditions under which a sequence of Riemannian manifolds with controlled curvature and injectivity radius admits a smoothly convergent subsequence.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Tarski–Mostowski–Robinson theorem Target entity description: The Tarski–Mostowski–Robinson theorem is a fundamental result in model theory that characterizes when a class of structures is first-order axiomatizable, linking definability properties with closure under ultraproducts and isomorphisms.
-
A.
Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
-
B.
Herbrand's theorem
Herbrand's theorem is a fundamental result in mathematical logic and proof theory that characterizes the validity of first-order formulas via finite sets of ground instances, forming a basis for automated theorem proving.
-
C.
Hilbert’s program
Hilbert’s program was an influential early-20th-century initiative in the foundations of mathematics that sought to formalize all of mathematics and prove its consistency using finitistic methods.
-
D.
Recherches sur la théorie de la démonstration
Recherches sur la théorie de la démonstration is Jacques Herbrand’s foundational work in mathematical logic, introducing key results in proof theory and what is now known as Herbrand’s theorem.
-
E.
Hamilton’s compactness theorem
Hamilton’s compactness theorem is a fundamental result in geometric analysis that provides conditions under which a sequence of Riemannian manifolds with controlled curvature and injectivity radius admits a smoothly convergent subsequence.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
theorem in mathematical logic
ⓘ
theorem in model theory ⓘ |
| appliesTo | classes of models of a first-order language ⓘ |
| assumes | fixed first-order signature ⓘ |
| characterizes | when a class of structures is first-order axiomatizable ⓘ |
| concerns |
classes of structures
ⓘ
definability in first-order logic ⓘ first-order axiomatizability ⓘ isomorphisms of structures ⓘ ultrapowers ⓘ ultraproducts ⓘ |
| context | classical first-order logic ⓘ |
| equates |
classes axiomatizable by first-order sentences
ⓘ
elementary classes ⓘ |
| field | model theory ⓘ |
| formalizes | link between ultraproduct closure and first-order definability ⓘ |
| givesConditionOn |
closure under isomorphisms
ⓘ
closure under ultraproducts ⓘ closure under ultraroots ⓘ |
| hasAlternativeName |
Tarski–Mostowski–Robinson theorem
ⓘ
surface form:
Tarski–Mostowski–Robinson characterization of elementary classes
|
| hasConsequence | non-elementary classes fail some closure property ⓘ |
| implies | closure properties of elementary classes ⓘ |
| isCentralResultIn |
axiomatizability theory
ⓘ
structural model theory ⓘ |
| isRelatedTo |
Robinson test for axiomatizability
ⓘ
compactness theorem ⓘ completeness theorem ⓘ Łoś–Tarski preservation theorem ⓘ |
| isUsedIn |
axiomatizability results in algebra
ⓘ
axiomatizability results in analysis ⓘ classification of model-theoretic classes ⓘ |
| namedAfter |
Abraham Robinson
ⓘ
Alfred Tarski ⓘ Andrzej Mostowski ⓘ |
| relates |
semantic closure properties
ⓘ
syntactic first-order definability ⓘ |
| statesThat | a class of structures is elementary if and only if it is closed under isomorphisms, ultraproducts, and ultraroots ⓘ |
| usesConcept |
elementary equivalence
ⓘ
first-order theory ⓘ ultrafilter ⓘ Łoś's theorem ⓘ |
| yearApprox | mid 20th century ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Tarski–Mostowski–Robinson theorem Description of subject: The Tarski–Mostowski–Robinson theorem is a fundamental result in model theory that characterizes when a class of structures is first-order axiomatizable, linking definability properties with closure under ultraproducts and isomorphisms.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Tarski–Mostowski–Robinson characterization of elementary classes